Question

find the equation of all horizontal asymptotes of the following function.
$f(x) = \frac{2}{x + 6} - 2$
answer attempt 2 out of 2
one horizontal asymptote
$-2$

Explanation:

Step1: Recall Horizontal Asymptote Rule

For a rational function \( y=\frac{a}{x - h}+k \), the horizontal asymptote is \( y = k \) (since as \( x\to\pm\infty \), \( \frac{a}{x - h}\to0 \)).

Step2: Analyze the Given Function

The function is \( f(x)=\frac{2}{x + 6}-2 \). Comparing with \( y=\frac{a}{x - h}+k \), we have \( k=-2 \). As \( x\to\pm\infty \), \( \frac{2}{x + 6}\to0 \), so \( f(x)\to0 - 2=-2 \).

Answer:

The equation of the horizontal asymptote is \( y = -2 \)